The Discrepancy of Random Rectangular Matrices

TL;DR

This paper provides a complete analysis of the discrepancy properties of random Bernoulli and Poisson matrices, advancing understanding of the Beck-Fiala conjecture for various matrix densities.

Contribution

It introduces novel probabilistic techniques to analyze discrepancy in sparse and dense random matrices, extending prior work to all densities.

Findings

Sharp discrepancy characterizations for Poisson matrices across aspect ratios

Extension of discrepancy results to sparse matrices

Application of second moment and Stein's methods to random matrices

Abstract

A recent approach to the Beck-Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoss (SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main tool is a conditional second moment method combined with Stein's method of exchangeable pairs. While previous approaches are limited to dense matrices, our techniques allow us to work with matrices of all densities. This may be of independent interest for other sparse random constraint satisfaction problems.